Few results exists for three dimensional domains in \(\mathbb {R}^n\)  or Riemannian manifolds; see [1]}, [2]}, [3]}, [4]}, [5]} and references therein. The additional difficulties introduced by higher dimension are highlighted in the introduction of [6]} and the challenges in geometry are outlined in [3]}. In the case when \(M\)  is a domain in \(\mathbb {R}^3\)  with Euclidean metric and \(\Gamma _{\varepsilon , a}\)  is a single small disk absorbing window, [2]}, [3]} gave an expansion for the average of the expected first arrival time, averaged over \(M\) , up to an unspecified \(O(1)\)  term.
These results were improved upon in [5]}, by using geometric microlocal analysis. Namely, the authors derived the bounded term and estimated the remaining term, moreover, they obtain these results for general Reimaniann manifolds. The case when \(\Gamma _{\varepsilon ,a}\)  is a small elliptic window was also addressed in [2]}, [3]}, [5]}. We also mention [14]}, where the author gave a short review of related works (up to 2012).
